Mandelbrot: Art, math, science, and works in progress

New York City exhibit explores how all these worlds collided in one brain.

The image above, generated from a relatively simple mathematical formula, has become iconic and permanently connected with the man who identified it: mathematician Benoit Mandelbrot. But its iconic nature has ended up leaving it an awkward no-man's land, dismissed by art critics as kitsch, and divorced from the underlying mathematics that generated it. Now, a small exhibit in New York City is attempting to place the Mandelbrot set and other mathematical constructs back in their original context: one that's part of a long history of visualizations playing a key part in the creative process of math and science.

Even though that sounds like a tall order, it's all handled in a small space on an upper floor of the Bard Graduate Center in Manhattan, where the exhibit will be on display until January. "The Islands of Benoit Mandelbrot" has been curated by Nina Samuel, a visiting professor who has a background in the histories of science and art.

Samuel told Ars that she was first attracted to the topic precisely because the images generated from Mandelbrot sets and their relatives have been derided as a cliché. What was supposed to have been a short research project ended up a PhD thesis. The exhibit is filled with works she's found in Mandelbrot's own papers, those of German mathematician Otto E. Rössler, and other material from Edward Lorenz held by the Library of Congress.

From pens to programmers

The works on display from the three mathematicians capture a time when visualization itself was changing dramatically. The oldest come from Lorenz, and it consists of simple line drawings on graph paper. Lorenz is famous for his role in the development of chaos theory, which was popularized through ideas like the butterfly effect. But Lorenz found that, while chaotic systems are extremely sensitive to initial conditions, many had a tendency to gravitate towards a limited set of conditions. For example: it's impossible to predict the weather in New York on a given July day, it's safe to expect that it will be warm.

These likely states are called "chaotic attractors," and the exhibit contains Lorenz' sketches of their three-dimensional representations, complex mixes of straight lines and curves, with dashed lines attempting to represent their three dimensionality.

Enlarge/ An attempt at representing a chaotic attractor in two dimensions.

J. Timmer

Rössler's work contained no shortage of sketches, though he tended towards colored pens and would sometimes perform revisions using bits of white tape to block over things that no longer satisfied him. But Rössler was working during the dawn of the computer age, and the exhibit shows off one of his early efforts: a simple visualization, done with an analog computer—but in stereo to attempt to capture the three-dimensionality of the underlying subject.

Mandelbrot did sketch things, but he was part of the first generation of mathematicians who relied heavily on computers. This was no small challenge, given that he didn't actually know how to program them himself. That's one of the reasons he started a long-term collaboration with IBM's Watson Research Center, where he had access to people who could turn his ideas into computer code, and high-end output devices to bring them to life. There's a wall full of that output on display in the exhibit, from a wall full of rough drafts of the classic Mandelbrot set to IBM ad copy of some artificial fractal landscapes.

Visualization as science and math

The fact that the need for visualization transcended a change in technology should probably speak to its central role. But both Samuel and the material she's gathered made that point explicit. Samuel described how some prints of Mandelbrot sets had small, individual pixels that might have been the result of an imperfect printing process. But, when zoomed in, each of these pixels represented a fractal world of its own. Not only did these visualizations reveal one of the central features of Mandelbrot sets, but they implied something about the mathematical system itself. Mandelbrot thought these speckles were islands, unconnected to some of the larger patterns around them and (erroneously, as it turned out).

(These disconnected dots were very important to him. In one book, the editor removed some of the apparent noise, assuming it was an artifact of the printing process. Samuel told Ars that, in copies he shared with colleagues, Mandelbrot actually added the speckles back by hand).

The visualizations also can form a key bridge between math and science. Mandelbrot also worked on patterns called Lévy flights, which describe paths where a random number of short hops are interrupted a single large leap. These Lévy flights create a very distinctive pattern when visualized (a number are on display in the exhibit). And, as people have studied foraging behavior in various animals, they've found that they also display the sort of series of hops and leaps typical of the Lévy flight. These parallels can be formally demonstrated mathematically, but the impetus for performing this demonstration was undoubtedly someone noticing the visual resemblance between the two patterns.

In Rössler's case, the link to science was explicit. There's a letter from him on display in which a series of hand sketches of three-dimensional surfaces are interspersed with typed notes. At some point, the text makes it clear that these complex shapes represent potential solutions to the Prisoner's Dilemma, a key example that's often used in game theory experiments.

Works in progress

The other thing that the exhibit makes clear is that the visualizations were part of the creative process. An entire wall is given over to a series of prints made from the famous Mandelbrot set. The zoom and location of the print relative to the entire image are varied in each one and, in many cases, it's a visual disaster—only a few random specks appear, with no pattern discernible. Samuel said these prints were like rough drafts, helping Mandelbrot understand what the equations he was working with said in various ways, and seeing how they varied in space and scale in a way that couldn't be done easily (if at all) by sketching on paper.

Enlarge/ Mandelbrot mixed computer output and hand-written notes when trying to come to grips with the math.

Collection of A. Mandelbrot.

Is any of it art? There's clearly a lot of technical skill behind the material, and a lot of thought and passion went in to producing some of it. With a few exceptions—like a dramatic, colored graph that ended up in the cover of Scientific American—it wasn't intentionally produced as art, but it's clear that's less of a barrier to things than it might otherwise be.

Since art is so subjective, it's probably best that anyone who wants to have that debate visit the exhibit itself. It's a small one, so it won't take long to digest the whole thing. But it could leave you considering the intersection between math, science, creativity, and visual arts for quite a bit longer than it takes to admire the displays.

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28 Reader Comments

The fact that the need for visualization transcended a change in technology should probably speak to its central role. But both Samuel and the material she's gathered made that point explicit. Samuel described how some prints of Mandelbrot sets had small, individual pixels that might have been the result of an imperfect printing process. But, when zoomed in, each of these pixels represented a fractal world of its own. Not only did these visualizations reveal one of the central features of Mandelbrot sets, but they implied something about the mathematical system itself. Mandelbrot thought these speckles were islands, unconnected to some of the larger patterns around them and (erroneously, as it turned out).

I wonder with the trend towards E-books, will this problem disappear?

Quote:

Since art is so subjective, it's probably best that anyone who wants to have that debate visit the exhibit itself. It's a small one, so it won't take long to digest the whole thing. But it could leave you considering the intersection between math, science, creativity, and visual arts for quite a bit longer than it takes to admire the displays.

The fact that the need for visualization transcended a change in technology should probably speak to its central role. But both Samuel and the material she's gathered made that point explicit. Samuel described how some prints of Mandelbrot sets had small, individual pixels that might have been the result of an imperfect printing process. But, when zoomed in, each of these pixels represented a fractal world of its own. Not only did these visualizations reveal one of the central features of Mandelbrot sets, but they implied something about the mathematical system itself. Mandelbrot thought these speckles were islands, unconnected to some of the larger patterns around them and (erroneously, as it turned out).

I wonder with the trend towards E-books, will this problem disappear?

The problem lays in the printing aspect, which is permanent and limited to pixel density. Resolving the image by zooming, whether in or out, on the fly or not, is not an issue with modern computers. This would include an ebook if a proper viewer were used.Example: http://neave.com/fractal/

Yep, mandelbulbs have come quite a long way. There's also IFS and flame fractals that have been well developed over the past decade. DeviantArt has a pretty large fractal art community, with a number of exceptional artists (including many of the developers of the most popular fractal generation programs).

There was a program on Discovery (or the like, can't remember exactly which channel right now) that I saw the other day (likely, it was a rerun since it was 'marathon' style... there were about four episodes being shown in a row) that talked about fractals, fairly interesting. It also talked about Jackson Pollock's art and how it resembled fractals.

It is hard to make rich art out of mathematics, because the emotional element is hard to convey. This artist avoids that problem by focusing on the people who did the maths.

It is quite a small audience that would understand the Mandelbrot set well enough to recognize an emotion that an artist was trying to transmit by using it in a meaningful way (not just as a "cool pattern").

But I think a theoretical understanding of complex dynamics could make it into art. I spend a lot of time trying to visualize extra dimensions in everyday life. No ideas myself on how to get the emotions that gives me into a painting or something, but it could be possible.

The fact that the need for visualization transcended a change in technology should probably speak to its central role. But both Samuel and the material she's gathered made that point explicit. Samuel described how some prints of Mandelbrot sets had small, individual pixels that might have been the result of an imperfect printing process. But, when zoomed in, each of these pixels represented a fractal world of its own. Not only did these visualizations reveal one of the central features of Mandelbrot sets, but they implied something about the mathematical system itself. Mandelbrot thought these speckles were islands, unconnected to some of the larger patterns around them and (erroneously, as it turned out).

I wonder with the trend towards E-books, will this problem disappear?

The problem lays in the printing aspect, which is permanent and limited to pixel density. Resolving the image by zooming, whether in or out, on the fly or not, is not an issue with modern computers. This would include an ebook if a proper viewer were used.Example: http://neave.com/fractal/

Well, displaying the Mandelbrot set is not such a problem anymore, but there's no standard for including code in an e-book. Your link leads to a Flash site, which is blocked by default here, and unavailable on most phones and tablets even though they have plenty of computing power for this. It would be surprising if the site still worked in 20 years. I don't know what a good choice is for writing an interactive graphical program that you would like to be widely usable in 30 years or so. HTML5 is probably the most universal now, but it doesn't scream stability.

"Mandelbrot did sketch things, but he was part of the first generation of mathematicians who relied heavily on computers."

I was in grad school - 1990-1995, when Benoit Mandelbrot first published his "findings". Sadly, the research mathematicians dismissed his work as "unoriginal" or not substantive enough. They said he was not introducing any new theory and that his work was more along the lines of stuff produced by Escher.The art community dismissed his work because it was not made by hand.

What is art? By definition, it is a form of creative expression that evokes an emotional response. Well, in my book the Mandelbrot sets do just that. Perhaps knowing the math behind the work helps, but Mandelbrot inspired a generation of recursive art, and new ways to look at bounded infinity.

Case in point: Recursion. 2003, by Elena Filippova.

You too can "make" a Mandelbrot-inspired design. Get some construction paper, or pages of a magazine, or even old photos that you don't mind cutting up. Cut out a simple shape, something that is symmetric about a single axis - like a leaf. Suggestion would be about 9" x 6", believe it or not. Note the dimensions and glue it to to the center of a background paper (24 x 24 inches). Note the "center" and "orientation" of the leaf. Also, using a separate paper, make an equilateral triangle pattern that has 3 sides each 18" in length.

Here's the recursion:Now, cut 3 leaves all 2/3 the size of the glued one but an exact copy, only miniaturized. With a pencil, lightly, mark the vertices of the equilateral triangle, with the glued leaf at the center. (To really make it interesting, rotate the triangle 45 degrees around that center point. Then, mark the 3 vertices.) Glue the 3 smaller leaves with center at each of these 3 vertices and orient them from the triangle's orientation. Now repeat for each of the 3 "glued" leaves as the center of each new set of 3, and a new triangle pattern, 2/3 the size of the previous one.

Note:In the second stage, you will cut 9 leaves in total. Third stage, 27, and so on. Also, note that the triangle will get smaller with each iteration. Triangle size and leaf size is proportional, so the second triangle pattern will be 12" each side.

Repeat at least 3 times. If you make it to a fourth stage, you will have 81 leaves, in addition to the previous 1 + 9 + 27.

That's a lot of construction paper - or magazine pages. But you have made your very own recursive, self-similar work of art. (And yes, you can use a computer. :-) )

I'm not sure whether the exhibit addresses it or not, but fractals were at the core of a very serious dispute within the philosophy of sciences in Europe.Here we then had the deterministic guys, classical way, and the ones opting for a scientific background for freedom and indeterminism.These latter ones gathered around the former Nobel prize I. Prigogine, who based his approach on a very precise account of complex physical systems. Most of those relied on attractors, thus: enters Mandelbrot.

I think Mandelbrot himself wasn't specially associating himself to scientific bases for indeterminism -indeed, among the fierce determinists, some also used attractors-like maths to try defending e. g. mathematical 'catastrophes' theories that definitely were deterministic (I remember a very serious paper called... 'halt to randomness, silence to noise!' which intended to negate the influence of noise, attractors, and more generally the 'butterfly effect' as it is now known).

But definitely, fractals as candidate dynamics system attractors were at the center of a real paradigm shift in Science, in other words: almost a war ;-)By these times having a fractal plot in your office was a blunt political position, no less: you were pro-, or you were anti-...

This too may explain why there were so many small-graphical-apps developments at the time, where on any computing platform you very quickly could find half a dozen "fractal generators", even though the "PC" at this time was the Apple II. Of course plots were in black and white in the beginning :-)

What I think is missed by many on the art community is the level of detail that emerges when zooming in on a Mandelbrot image. Think of something like Brueghel's Fall of Icarus or any of the interior scenes done by Vermeer. There is ridiculous amounts of detail that can be seen only by "zooming in" on the painting. (In these cases, zooming in is done photographically by taking a picture and enlarging it.)

If I had to guess as to why the art community looks down on mathematically-created art it is because there is not much agency in the creation of these artworks. Mandelbrot didn't create the patterns named after him, he discovered the formulas that allow them to be created automatically. As the article points out, it was the visualization of the patterns that helped Mandelbrot refine his formulas, so it is the formulas that "create" the art and not Mandelbrot himself. (Although personally, I find a lot of appeal in the patterns and symmetry that can be found in Mandelbrot images. It's like a lot of modern sculpture: abstract at first glance but usually imbued with an underlying pattern on closer examination.)

Edited to add the following.

Look at Brancusi's Bird in Space sculpture. It was done all by hand but nowadays a mathematician could probably devise a formula to create a similarly shaped object.

I once (ONCE) generated that image on the display of... a <b>Commodore 64</b>, after reading Douglas Hofstadter's initial article in Scientific American. It was monochrome, and it took hours to render. But when I came home later, there it was. Beeyootiful, in a BASIC-yet-complex way (yes I know I should be shot for that).

But the point is: <b>a Commodore 64!!!</b> Can I be upgraded from Smack-Fu Master on the strength of that? Or did I ruin my chances with those puns?

I once (ONCE) generated that image on the display of... a Commodore 64, after reading Douglas Hofstadter's initial article in Scientific American. It was monochrome, and it took hours to render. But when I came home later, there it was. Beeyootiful, in a BASIC-yet-complex way (yes I know I should be shot for that).

But the point is: a Commodore 64!!! Can I be upgraded from Smack-Fu Master on the strength of that? Or did I ruin my chances with those puns? (and formatting failure).

I once (ONCE) generated that image on the display of... a Commodore 64, after reading Douglas Hofstadter's initial article in Scientific American. It was monochrome, and it took hours to render. But when I came home later, there it was. Beeyootiful, in a BASIC-yet-complex way (yes I know I should be shot for that).

Nice times indeed. I right away went for assembly and colors, despite those funny 2:1 pixels - only maybe 4 times faster though since floating point was emulated with the same ROM routines as in BASIC. A perfect small project for sure, does Sci Am still have these occasionally? I even feel an urge now to try it on today's hardware.

Is it art? When it comes to choosing region of the plane and coloring scheme, perhaps. I remember all the frustration trying to come up with something appealing to the eye. Or is that just graphic design

Great article over all but two sticking points. Why no mention of Hubbard and Douady, the men who actually started the mathematical study of the Mandelbrot set? And secondly, what is with the nonsense about the disconnected specks? Mandelbrot was a mathematician. He knew plenty well (from the work of Douady and Hubbard) that the Mandelbrot set is connected and conformal to the unit disk.

Edit: It seems that I should check my dates about these things. Mandelbrot conjectured it was disconnected before the results of Hubbard and Douady were published.

My understanding of the Topology of the "boundary" of the Mandelbrot Set is that it is *not*, in fact homeomorphic to the unit circle. In other words, in layman's terms, you can't start walking along the boundary and travel the entirety returning to one's starting point. To quote Wolfram Alpha:"Shishikura (1994) proved that the boundary of the Mandelbrot set is a fractal with Hausdorff dimension 2, refuting the conclusion of Elenbogen and Kaeding (1989) that it is not. However, it is not yet known if the Mandelbrot set is pathwise-connected. If it is pathwise-connected, then Hubbard and Douady's proof implies that the Mandelbrot set is the image of a circle and can be constructed from a disk by collapsing certain arcs in the interior (Douady 1986)."

In other words, it is still an open question as to the pathwise connectedness of the set itself. This is a requirement for the Mandelbrot Set to be homeomorphic to a disk. Personally, it is my belief that it is not. Particularly because of the nature of the boundary.

I was in grad school - 1990-1995, when Benoit Mandelbrot first published his "findings".

The Fractal Geometry of Nature was published in 1977. His commodities studies (economics info had the best data set to work with) were published back in the sixties. So I really don't know which air-quoted findings you are talking about.

I was in grad school - 1990-1995, when Benoit Mandelbrot first published his "findings".

The Fractal Geometry of Nature was published in 1977. His commodities studies (economics info had the best data set to work with) were published back in the sixties. So I really don't know which air-quoted findings you are talking about.

Yes I should clarify. It wasn't until 1999 that Prof Mandelbrot achieved a tenured position. Before this time, he had managed to publish his works in various mainstream publications. But as it was explained to me, there were few research journals in mathematics who would give his work serious consideration. I'm referring to the work beyond the initial discovery of these fractals (sets with fractional dimension), his use of computer modeling to delve more into the secrets of the boundary of this set.

So, at the time I was in grad school, privately, mathematicians were admiring the beauty of his work but were voicing disappointment in the rigor (or lack thereof) of the mathematical substance. Sad, really. Over the years, this attitude has changed.

My understanding of the Topology of the "boundary" of the Mandelbrot Set is that it is *not*, in fact homeomorphic to the unit circle. In other words, in layman's terms, you can't start walking along the boundary and travel the entirety returning to one's starting point. To quote Wolfram Alpha:"Shishikura (1994) proved that the boundary of the Mandelbrot set is a fractal with Hausdorff dimension 2, refuting the conclusion of Elenbogen and Kaeding (1989) that it is not. However, it is not yet known if the Mandelbrot set is pathwise-connected. If it is pathwise-connected, then Hubbard and Douady's proof implies that the Mandelbrot set is the image of a circle and can be constructed from a disk by collapsing certain arcs in the interior (Douady 1986)."

In other words, it is still an open question as to the pathwise connectedness of the set itself. This is a requirement for the Mandelbrot Set to be homeomorphic to a disk. Personally, it is my belief that it is not. Particularly because of the nature of the boundary.

I have deep respect for algebraic topologists - some of the stuff is so out there and "pathological", that it boggles the mind. OT, science and engineering majors would get a way better introduction to rigorous, abstract reasoning by taking Real Analysis or Introduction to Topology instead of Calc x00.